EUCLIDEAN ASYMPTOTIC EXPANSIONS OF GREEN FUNCTIONS

arXiv:hep-ph/9612287v1 8 Dec 1996

FERMILAB-PUB-91/345-T

EUCLIDEAN ASYMPTOTIC EXPANSIONS OF GREEN FUNCTIONS OF QUANTUM FIELDS. (II) Combinatorics of the asymptotic operation G. B. Pivovarov Institute for Nuclear Research of the Russian Academy of Sciences, Moscow 117312, Russia. and F. V. Tkachov1 Fermi National Accelerator Laboratory P.O.Box 500, Batavia, Illinois 60510 USA

ABSTRACT The results of [1] are used to obtain full asymptotic expansions of Feynman diagrams renormalized within the MS-scheme in the regimes when some of the masses and external momenta are large with respect to the others. The large momenta are Euclidean, and the expanded diagrams are regarded as distributions with respect to them. The small masses may be equal to zero. The As-operation for integrals is defined and a simple combinatorial techniques is developed to study its exponentiation. The Asoperation is used to obtain the corresponding expansions of arbitrary Green functions. Such expansions generalize and improve upon the well-known short-distance operator-product expansions, the decoupling theorem etc.; e.g. the low-energy effective Lagrangians are obtained to all orders of the inverse heavy mass. The obtained expansions possess the property of perfect factorization of large and small parameters, which is essential for meaningful applications to phenomenology. As an auxiliary tool, the inversion of the R-operation is constructed. The results are valid for arbitrary QFT models. 1

on leave from Institute for Nuclear Research of the Russian Academy of Sciences, Moscow 117312, Russia

2

1

Introduction

In the preceding paper [1] the problem of obtaining asymptotic expansions of multiloop Feynman diagrams was analyzed. From the point of view of phenomenology, it was stressed that such expansions must possess the property of “perfect factorization” of large and small dimensional parameters. From the point of view of mathematics, it was shown that the key technical problem (“the Master problem”) is that of obtaining asymptotic expansions—in powers and logarithms of the expansions parameter—of products of singular functions in the sense of distributions. An explicit solution for the Master problem was obtained in the form of the so-called As-operation for products of singular functions, which acts on Euclidean momentum-space Feynman integrands and yields an expansion whose all terms are well-defined distributions (containing, in particular, δ-functional contributions with coefficients depending non-analytically on the expansion parameter) so that the expansion allows termwise integrations with test functions. The purpose of the present paper is to use the results of [1] and obtain explicit expansions of multiloop Feynman integrals in the form of As-operation for integrals.1 It turns out that the combinatorial structure of the As-operation for integrals is very similar to that of the R-operation2 . This similarity allows one to derive expansions of perturbative Green functions in a global form with the help of the same techniques as used in studying the exponentiation of the R-operation. The present paper is a revised and simplified version of our publications [2], [3] and [4] which have been widely discussed in the literature [20]–[22].3 The revision has not affected the results but only the order of presentation. The simplification is due to the fact that the starting point of [2] was the so-called EA-expansion for UV-renormalized integrals, which is a composition of R-operation and the As-operation in the sense of the present paper. Correspondingly, the As-operation as defined in [2] mixed up UV-renormalization and the expansion algorithm. In the present version we define the As-operation and study its exponentiation directly for the unrenormalized integrals, which results in further simplifications. The attractiveness of the As-operation is due to the fact that it serves as a combinatorial organizing center of the theory and fully exhibits the structure of the expansions, which is invaluable for applications where diagram-by-diagram analysis is necessary to enhance reliability of calculations. Its definition is also remarkably simple within dimensional regularization, which trivializes the study of exponentiation. It should be stressed, however, that the use of dimensional regularization is by no means essential for the expansion problem; a separate publication is devoted to a study of this point—cf. [14], [15], and especially [17]. In the latter work Euclidean asymptotic expansions were studied in regularization indepen1

The two As-operations are different representations of essentially the same operation, which justifies using the same name to denote them: the As-operation for integrals is an integrated version of the As-operation for products—cf. subsect. 5.1 below. 2 Note that the As-operation for products also has close parallels with the R-operation—but with the R-operation in position representation. 3 OPE in the MS-scheme was also studied in [19].

1 INTRODUCTION

3

dent manner within the framework of generalized MS-schemes [15]. The main result of [17] is an expansion formula whose combinatorial structure is practically identical to that of the As-operation of the present paper. Therefore, all our combinatorial results concerning exponentiation of the As-operation are immediately applicable to [17]. The plan of the present paper is as follows. It consists of five sections, each section is subdivided into subsections and contains a prologue where further information can be found. In the first three sections various aspects of combinatorial structure of the R-operation in the MS-scheme are studied. In sect. 2 the exponentiation formula for the R-operation is obtained using the functional techniques. The result is, of course, well-known. However, since we wish to use similar arguments in studying the Asoperation, we feel it is useful to present an explicit derivation in such a way as to make it immediately applicable to the case of the As-operation studied in sects. 4 and 5. In sect. 3 the inversion of the R-operation is derived, and in sect. 4 the renormalization of Green functions with multiple local operator insertions is considered. In sects. 4 and 5 we study the combinatorial structure of asymptotic expansions of Feynman diagrams. In sect. 5 we consider the special case of heavy mass expansions. The As-operation for integrals is introduced and its exponentiation is proved. Using the techniques developed in sects. 1–3, the effective low-energy Lagrangian is presented in an explicitly convergent form to all orders in the inverse heavy mass. (It should be stressed that in this paper we only deal with purely combinatorial aspects of the theory. A fuller justification of a technical assumption (see subsect. 5.8 below) is presented in [17] (for an informal discussion see [16].) In sect. 6 the results of sect. 5 are extended to comprise the case of large external momenta, so that the familiar short-distance operator product expansion is reproduced and its generalizations are obtained. Note that our final formulae immediately provide the very useful explicit expressions of OPE coefficient functions first announced in [11] (explicit examples of calculations going as far as 3-loop approximation can be found in [18]; for further references see also [23]). The notations are on the whole consistent with those of [1].

4

COMBINATORICS OF THE R-OPERATION IN THE MS-SCHEME

2

R-operation in the MS-scheme

In this section we study the combinatorial structure of the R-operation applied to the perturbation series as a whole. The fact that the R-operation is equivalent to adding counterterms to the Lagrangian is of course well-known. But we are interested in exact formulae expressing this fact that would be convenient for practical calculations and valid for the MS-scheme of UV renormalization. Therefore we reproduce a simple proof of the exponentiation of the R-operation in the MS-scheme—see (2.25) below—as given in [2]. Our second aim is to present the derivation of the exponentiation formula (2.25) in a form sufficiently general to make it applicable to the case of the As-operation studied in sects. 4 and 5. To make the paper more accessible to practitioners of applied QFT, our presentation is rather detailed. We start in subsect. 2.1 with a description of the R-operation in the MS-scheme. Our definitions are equivalent to the standard ones [5], and the differences are mainly notational. In subsect. 2.2 a convenient order of enumeration of UV-subgraphs is fixed in preparation to studying the global structure of the R-operation. Subsect. 2.3 summarizes some results of the functional techniques that will be needed (for more details see e.g. [7]). In subsect. 2.4 a formal expression of the statement that the Roperation is equivalent to adding counterterms to the Lagrangian, the exponentiation formula (2.25), is derived; our reasoning is an extension of the arguments of [6]—for comments and comparison see subsect. 2.4). In subsect. 2.5 the results are extended to Green functions of fields. Complications due to composite operator insertions will be discussed in sect. 4.

2.1

Basic definition of the R-operation

The effect of the R-operation in the MS-scheme on a single Feynman integral G can be expressed by the formula R◦G =

X

{gα }

Y α

!

∆UV ◦gα × G/

Y α

!

gα .

(2.1)

Let us explain it. One associates a Feynman graph, that will be referred to as “the graph G”, with the integral G, and vice versa. The correspondence between graphs and integrals is established by Feynman rules. Let us fix how µ, the ’t Hooft unit of mass, appears in the integrals. In momentum representation, having taken into account momentum conservation in all the vertices of

2 R-OPERATION IN THE MS-SCHEME

5

the graph, one can obtain the integrand of the corresponding unrenormalized Feynman integral. Then for each internal momentum integration one replaces the measure as D d4 p 2ǫ d p → µ , (2π)4 (2π)D

2ǫ = 4 − D,

(2.2)

and assumes that integrations (as well as the Dirac algebra etc.) are to be done within dimensional regularization. The role of µ is to preserve the dimensionality of the integral as a whole under regularization. An UV-subgraph g of G consists of some vertices of the graph G and some of its lines attached by both ends to the vertices of g. If g consists of one vertex and no lines then we call it elementary vertex. If g has at least one loop and is connected and one-particle-irreducible then we call g a proper UV-subgraph. (Note that if G is one-particle-irreducible then one of its proper UV-subgraphs coincides with G.) An UV-partition of G is a set of its UV-subgraphs {gα } such that each vertex of G belongs to one and only one gα . Note that one of the UV-partitions consists of elementary vertices only, and another one consists of only one subgraph, namely, G as a whole. Summation in (2.1) runs over all UV-partitions of G. Now we have to define the operation ∆UV ◦g where g is an UV-subgraph of G. By definition, ∆UV ◦g = g if g is an elementary vertex, and ∆UV ◦g = 0 if g is not an UVsubgraph. If g is a proper UV-subgraph, then the action of ∆UV on g can be described as follows. The graph for (∆UV ◦g) × (G/g) can be obtained from the graph for G by shrinking g in G to a point. With the new vertex thus obtained one associates a factor—the UV-counterterm—that is obtained from the integral corresponding to g via a special algorithm which need not be specified here (see, however, below). Using the Feynman rules thus extended one finally builds up the integral for (∆UV ◦g) × (G/g). Note that the above rule for µ is still operative, i.e. the number of the factors µ2ǫ is equal to the number of loops of G with g shrunk to a point. For the sake of completeness let us describe the recipe for evaluating the UVcounterterm for a given proper UV-subgraph g. If one rewrites the R-operation as R◦g ≡ R′ ◦g + ∆UV ◦g,

(2.3)

then ∆UV ◦g is precisely the UV-counterterm to be evaluated while R′ involves only counterterms for subgraphs of g which are assumed to be known already. Since we are working within dimensional regularization, all our integrals depend parametrically on D, the complex-valued dimension of space-time. Let K be the operation that picks out the pole part at D = 4 of any function on which it acts. R◦g should be finite at D = 4, therefore K◦R◦g = 0. On the other hand, the UV-counterterms in the MS-scheme are pure poles. So, applying K to (2.3), we get: ∆UV ◦g = −K◦R′ ◦g.

(2.4)

Eqs. (2.1), (2.3) and (2.4) provide a convenient description of the R-operation in the MS-scheme. We would like to stress that all the manipulations are to be done in


6

momentum representation, and that the rule for µ remains operative. Also recall that (2.4) is a polynomial of masses and external momenta of g and is independent of µ. The most important points for us here are: (i) ∆UV ◦g, whatever the subgraph g is, depends only on g but not on G as a whole. (ii) Summation in (2.1) runs over all partitions of G. (iii) An UV-subgraph is defined as a subset of vertices and some (not necessarily all) of the lines connecting the vertices of this subset. On the other hand, the fact that ∆UV is non-trivial only on proper UV-subgraphs will be completely irrelevant.

2.2

Enumerating UV-subgraphs

Let us transform (2.1) to a more convenient form. Indeed, there are UV-subgraphs that have the same set of vertices but differ only with respect of the number and/or arrangement of lines connecting them (e.g. when a line of a subgraph can be removed without changing its status as 1PI)—but not vice versa: if the sets of lines of two subgraphs are identical, than their sets of vertices are identical, too. This allows one to perfrom enumeration of UV-subgraphs in (2.1) in two stages: first, one enumerates sets of vertices, then, sets of lines connecting those vertices. Indeed, let v be a subset of vertices of G and Λ v ◦G ≡

X

(∆UV ◦g) × (G/g),

(2.5)

g∼v

where summation runs over all UV-subgraphs g with the same set of vertices, v. Denoting the set of all vertices of G as V , we can rewrite (2.1) as: R◦G =

X

v=∪vα vα ∩vβ =∅, for α6=β

2.3

Y α

Λvα

!

◦

G.

(2.6)

T -products and functional techniques

Let ϕ(x) ˆ be a free field operator; ϕ(x) will denote the corresponding classical field. Let F (ϕ) be a functional of the form F (ϕ) =

XZ

Fn (x1 . . . xn )ϕ(x1 ) . . . ϕ(xn )dx1 . . . dxn .

(2.7)

n

Replacing the product of ϕ’s in (2.7) by the T -product of ϕ’s ˆ one obtains an operator which we denote as TF (ϕ). ˆ The Wick theorem allows one to reexpand TF (ϕ) ˆ in terms of the Wick normal products of ϕˆ : TF (ϕ) ˆ =

XZ n

FnN (x1 . . . xn ) N [ϕ(x ˆ 1 ) . . . ϕ(x ˆ n )] dx1 . . . dxn ,

(2.8)


7

where FnN are related to Fn in a certain way (see below). The generating functional for the coefficient functions is defined as: F N (ϕ) =

XZ

FnN (x1 . . . xn )ϕ(x1 ) . . . ϕ(xn )dx1 . . . dxn .

(2.9)

n

The relation between F (ϕ) and F N (ϕ) is expressed, by definition, by TF (ϕ) ˆ = N[F N (ϕ)]ϕ=ϕˆ .

(2.10)

The well-known formal expression of the Wick theorem due to Khori (see [7] and refs. therein) reads: F N (ϕ) = el F (ϕ), (2.11) where

1 l≡ 2

Z

dx dy

δ δ 1 ∆c (x, y) ≡ δ∆c δ, ϕ(x) ϕ(y) 2

(2.12)

and ∆c (x, y) =< Tϕ(x) ˆ ϕ(y) ˆ >0 .

(2.13)

(Extension of all our formulae to the most general case of fermionic and complex fields is straightforward, therefore our analysis will be quite general.) Our purpose is to study Green functions, but first it is convenient to consider the (off-shell) S-matrix. The unrenormalized S-matrix is expressed as: S(ϕ) ˆ = T exp[L(ϕ)], ˆ

(2.14)

where L(ϕ) ˆ denotes the interaction Lagrangian integrated over space-time and multiplied by i (the normal ordering of fields in L(ϕ) ˆ is not assumed). The generating functional S N is defined as (cf. (2.10)): T exp[L(ϕ)] ˆ = N[S N (ϕ)]ϕ=ϕˆ

(2.15)

S N = el exp[L(ϕ)].

(2.16)

and from (2.11): Again we have to set the rules for µ, in order to establish connection between our earlier definitions of Feynman integrals and the expressions generated using the functional technique. We assume that: (i) all momentum and space-time integrations, γ-matrices etc. are D-dimensional; (ii) the dimensions of the fields ϕˆ and the coupling constants are always the canonical 4-dimensional ones; (iii) each propagator bears an extra factor µ2ǫ ; note that (2.12) is dimensionless since we assume that δ/δϕ(x) × ϕ(y) = δ(x − y) with D-dimensional δ-function; (iv) the interaction Lagrangian L(ϕ) in (2.14) contains the factor µ−2ǫ and is therefore dimensionless; note that condition (iii) is equivalent to (iv) extended to the quadratic part of the full Lagrangian ( cf. (2.35) below).


8

Given these rules, a connected Feynman integral will have the factor µ2ǫ(l−1) where l is the number of its internal momentum integrations (loops)—which differs by µ−2ǫ from what was postulated in subsect. 2.1. Such extra µ−2ǫ to each connected component can be easily taken into account in what follows.

2.4

Exponentiation of the R-operation

The renormalized S-matrix is obtained by applying the R-operation to each Feynman integral contributing to the coefficient functions of S N . Thus, the starting point of our analysis is the following expression: SR (ϕ) = R [T exp(L(ϕ))] = ◦

∞ X

1 R◦TLN (ϕ), N =0 N!

(2.17)

where we have introduced the convenient notation T ≡ el ,

(2.18)

which will be used systematically in the context of the functional techniques. Evaluating functional derivatives, one obtains a sum of terms, each one having a graphic representation. In the resulting expression one only has to replace products of the classical fields ϕ by the normal (N-) products of the free fields ϕ, ˆ in order to obtain the operator of the S-matrix. (Note that since we wish to use the MS-scheme for UV renormalization, the S-matrix will not satisfy correct normalization conditions automatically, but this is of no importance because our study of the S-matrix is only an intermediate step in the study of Green functions.) Note that in (2.17) each L(ϕ) corresponds to a vertex while each l (see (2.12)) generates one line of a Feynman graph. This allows one to conveniently perform all the enumerations of UV-partitions inherent in the definition of the R-operation as given by eqs. (2.1)–(2.4). First one has to enumerate all possible partitions of N vertices into groups of vertices of various sizes (cf. (2.6)). N vertices can be split into n1 groups of 1 vertex, n2 groups of 2 vertices . . . nN groups of N vertices in (

N n1 . . . nN

)

= N!

"

N Y

nk !(k!)

nk

#−1

,

X

k × nk = N,

(2.19)

n1 nN N Λ◦L1 . . . Λ◦LN . n1 . . . nN

(2.20)

k=1

k

ways. Therefore, applying (2.6) to (2.17) we get: ∞ X

X 1 T n1 ...nN N =0 N! P

knk =N

(

)

(Note the abuse of notation: Λ acts on subgraphs consisting of vertices and lines—but the latter will only be generated by T. However, we use such a representation only for


9

the purposes of combinatorial enumeration so that no problems should arise.) Now, for each group of vertices we have to expand Λ according to (2.5). We have to separate the lines connecting vertices inside each group Λ◦Lk from the lines connecting vertices of different groups, and among the former to enumerate all subsets of lines that will form UV-subgraphs on which ∆UV acts. This can be done by ascribing different labels to the n1 + . . . + nN groups of vertices and marking the field ϕ in each group by this label. Then we have: T

" Y α

"

#

Y 1 Λ◦Lkα (ϕα ) Λ◦Lkα (ϕ) = exp( δ∆c δ) 2 α









= exp  = exp 

X

δα ∆c δβ +

X α

α0 ,

(4.1)

where ˜jk are Fourier transformed local products of ϕ(x) and its derivatives as defined in (3.1). Using the formalism introduced in the preceding section, one gets: δ δ ... R◦T eL δLnN (pN ) δLn1 (p1 )

eq.(4.1) =

=T =T

("

δ δ ... eξ[L] δLnN (pN ) δLn1 (p1 )

−ξ[L]

e

!

L=L, ϕ=0

!

δ δ ... eξ[L] δLnN (pN ) δLn1 (p1 )

L=L, ϕ=0

#

ξ[L]

e

L=L

)

.

(4.2)

ϕ=0

The expression in the square brackets on the r.h.s. of (4.2) is a sum of products of terms like "

#

i h δ δ ... ξ[L] = ∆◦ T ˜ji1 (p) . . . ˜jin (pn ) exp L . δLin (pn ) δLi1 (p1 )

(4.3)

It is not difficult to see that (4.3) is a local functional which can be expanded in ˜j as: eq.(4.3) =

X

˜jm (p1 + . . . + pn ) Z˜i(n)...i ;m (p1 . . . pn ), n 1

(4.4)

m

where Z˜ are polynomials in p and in the parameters of L. Z is independent of p if the dimensionality of ˜j in units of mass is greater than that of the product ˜ji1 . . . ˜jin . Moreover, since we include full derivatives into j, Z can not contain factors depending only on p1 + . . . + pn ; in particular, Z˜n(1) (p) are independent of p. Examples will help to understand the above. Denoting (4.3) as ∆(j1 . . . jn ), one has (LR was defined in (2.26)): R◦ < T j1 eL >0 = (T ∆(j1 ) exp[LR ])ϕ=0 ,

(4.5)

R◦ < T j1 j2 eL >0 = (T [∆(j1 )∆(j2 ) + ∆(j1 j2 )] exp[LR ])ϕ=0 ,

(4.6)

ˇ 3) R◦ < T j1 j2 j3 eL >0 = (T [∆(j1 )∆(j2 )∆(j3 ) + ∆(j1 j2 )[∆(j ˇ 1 j3 )∆(j ˇ 2 ) + ∆(j ˇ 2 j3 )∆(j ˇ 1 ) + ∆(j ˇ 1 j2 j3 )] exp[LR ])ϕ=0 . + ∆(j

(4.7)

4 RENORMALIZATION OF MULTILOCAL OPERATOR INSERTIONS

19

ˇ ˇ Since ∆(ϕ) = ϕ and ∆(ϕj 1 . . . jk ) = 0 (this is because it is impossible to form a 1PI graph with one “ϕ-insertion”), one can see that for j = ϕ the above formulae degenerate into R◦ < T ϕ(q ˜ 1 ) . . . ϕ(q ˜ n )eL >0 = (T ϕ(q ˜ 1 ) . . . ϕ(q ˜ n )eLR (ϕ) )ϕ=0 ,

(4.8)

which agrees with (2.33), as expected. Using (4.4), one can represent (4.5) as: R◦(T ˜j1 (p)eL ) =

X

(1) Z˜1;m (T ˜jm (p) exp[LR ])

(4.9)

m

(identities of this kind were first obtained by Zimmermann [9]), R◦(T ˜j1 (p1 )˜j2 (p2 )eL ) =

X

(2) Z˜1,2;m (p1 , p2 )(T ˜jm (p1 + p2 ) exp[LR ])

m

+

X

(1) Z˜1;m

m

X

(1) Z˜1;n (T ˜jm (p1 )˜jn (p2 ) exp[LR ]),

(4.10)

n

etc. Note that all the sums in (4.9) and (4.10) are finite. Identities like (4.10) and their generalizations can be called (generalized) Zimmermann identities for multilocal operator insertions.

4.2

Inverted Zimmermann identities

The equations derived in the preceding subsection allow one to express renormalized Green functions in terms of unrenormalized ones. Using the inversion of the Roperation obtained in sect. 3, one can easily do the opposite, i.e. reexpress unrenormalized quantities in terms of renormalized ones. Thus, instead of (4.1)–(4.2) one has: < T ˜j1 (q1 ) . . . ˜jn (qn )eL >0 δ δ ... T eL δLnN (pN ) δLn1 (p1 )

= = R ◦T

("

¯ −ξ[L]

e

!

L=L, ϕ=0

δ δ ¯ ... eξ[L] δLnN (pN ) δLn1 (p1 )

#

¯ ξ[L]

e L=L

)

,

(4.11)

ϕ=0

and instead of (4.9) one has: T ˜j1 (p) eL =

X

(1)

˜˜ R◦(T ˜j (p) exp ξ −1 [L]) Z m 1;m

m

˜˜ (1) = [Z˜ (1) ]−1 ), (note that Z T ˜j1 (p1 )˜j2 (p2 )eL =

X m

˜˜ (2) (p , p )R◦(T ˜j (p + p ) exp ξ −1[L]) Z m 1 2 1,2;m 1 2

(4.12)

5 HEAVY MASS EXPANSIONS

+

X m

˜˜ (1) R◦(T ˜j (p )˜j (p ) exp ξ −1 [L]), ˜˜ (1) X Z Z m 1 n 2 1;n 1;m

20 (4.13)

n

˜˜ (2) (p , p ) = − P Z˜ (2) (p , p )Z ˜˜ (1) ). And so on. (one can check that Z l 1,2;l 1 2 1,2;m 1 2 l;m

In a similar manner one can relate expressions renormalized in different renormalization schemes, cf. subsect. 3.7.

AS-OPERATION FOR MULTILOOP INTEGRALS.

5

Heavy mass expansions

We now turn to asymptotic expansions of Feynman diagrams. In this section we consider the case when the set of the heavy parameters contains only masses, which corresponds to studying low-energy effective Lagrangians to all orders in the inverse heavy mass. The reasoning in the more general case is very much the same, so that in the next section we will be able to concentrate only on specific complications due to large external momenta. In subsect. 5.1 the As-operation for unrenormalized (or UV-convergent) integrals is defined as an integrated version of the As-operation for products of singular functions introduced in [1], in subsect. 5.2 the combinatorial structure of the corresponding subgraphs is studied. In subsect. 5.3 the δ-functions corresponding to the counterterms for the IR-subgraphs are integrated out and in subsect. 5.4 the final form of the Asoperation for 1PI integrals is presented. In subsect. 5.5 the obtained expression for the As-operation is extended to non-1PI integrals, in subsect. 5.6 its exponentiation is proved, and in subsect. 5.7 it is transformed to an explicitly convergent form using the inversion of the R-operation. In subsects. 4.8 and 4.9 the results are extended to UV-renormalized diagrams and models, respectively. Explicit expressions for lowenergy effective Lagrangians are presented to all orders of the inverse heavy mass. Subsect. 5.10 contains a simple example.

5.1

As-operation for integrals: definition

Let us introduce notations and formulate the problem that we are going to consider here. Our notations on the whole will be consistent with subsect. 3.1 of [1]. Let Γ be an l-loop 1PI Feynman diagram, and let p = (p1 . . . pl ) denote its integration momenta. We assume in this section that Γ depends on the light and heavy masses m and M, and that all the external momenta of Γ are light (and denoted as k). All M are non-zero and independent of the expansion parameter denoted as κ, while

21


m, k = O(κ) and some masses m may be equal to zero. Unlike the abstract analysis of [1], here it will be convenient to show the dependence on m, k and M explicitly. Thus, the unrenormalized integrand is denoted as Γ(p, m, k, M). We assume that Γ is UV-convergent, i.e. Γ(p, . . .) is absolutely integrable over p in infinite limits. Denote the integrated diagram as: def

Γ(k, m, M) =

Z

dp Γ(p, k, m, M).

(5.1)

In [1] we introduced the As-operation for products of singular functions like Γ(p, k, m, M) which yields asymptotic expansions of such products with respect to the light parameters in the sense of the distribution theory. It is natural to expect (for more details see subsect. 5.8; a fuller justification is given in [17]) that if (5.1) is UV-convergent, then the asymptotic expansion of Γ(k, m, M) can be obtained by expanding its integrand Γ(p, k, m, M) using the As-operation of [1] and then performing termwise integration of the resulting series over p. Let us introduce an operation that acts on integrated Feynman diagrams in accordance with the above recipe. This new operation will be referred to as the As-operation for integrals. To distinguish the two types of As-operations, one could use e.g. the notations Asprod and Asint . But as it is normally clear whether the As-operation acts on a product of singular functions or on an integral, we will use the same notation As without subscripts in both cases. Formally, one has: def

As Γ(k, m, M) = ◦

Z

dp As◦Γ(p, k, m, M).

(5.2)

Let us exhibit the structure of the As-operation on the r.h.s. of (5.2). The general expression for it was given in [1]. Specifying it to the case considered here, one obtains: As◦Γ(p, k, m, M) = Tm,k ◦Γ(p, k, m, M) +

X

[E◦γ(p, k, m)] × [Tm.k ◦Γ\γ(p, k, m, M)],

(5.3)

γ

where the notations used are as follows. The operation Tm,k performs the Taylor expansion in powers of the light parameters. Summation runs over all IR-subgraphs of Γ. Specifying to the present case the definitions of subsects. 9.2–3 of [1], one sees that an IR-subgraph γ can be described as follows: (i) the set of lines of γ is a full subset of the light lines of Γ, i.e. if one puts to zero the momenta flowing through all the lines of γ together with the light external momenta k, then no light line which does not belong to γ should have the momentum flowing through it nullified owing to momentum conservation (cf. [1]); (ii) any vertex of Γ whose incident lines all belong to γ, is included into γ. The product of the lines and vertices of γ is independent of M and is denoted as γ(p, k, m). The product Γ\γ in (5.3) comprises all the factors from Γ that do not belong to γ. Graphically, Γ\γ is obtained from Γ by deleting all the lines and vertices of γ.

22


The operation E (expressions for it were derived in [1]) applied to an IR subgraph returns a linear combination of counterterms that are proportional to δ-functions of p. Our first aim here is to integrate out those δ-functions explicitly.

5.2

Structure of Γ\γ.

It is not difficult to verify that in the case under consideration (no heavy external momenta) the IR-subgraphs are uniquely characterized by the properties of their complements. Indeed, denote the connected components of Γ\γ as hi . These subgraphs (which will be referred to as heavy knots) have the following properties: (i) since γ consists of only light lines, each hi has only light external lines while all the heavy lines of Γ are hidden within all hi ; (ii) since γ is full, none of hi can have a light line whose momentum vanishes due to momentum conservation when one puts to zero all the external momenta of hi . This is an analytical formulation. Graphically, it implies that each hi is “1PI with respect to light lines” which means that it cannot be divided into two disconnected graphs (which may consist of a single vertex) by cutting any one of its light lines. Note that a heavy knot hi may consist of a single heavy line (including both vertices to which it is attached). One has: Γ\γ =

Y

hi .

(5.4)

i

In the final form of the As-operation for integrals, the heavy knots will play the same role as UV-subgraphs in the expression (2.1) for the R-operation. Example. Consider the diagram Γ as shown in Fig. 1a. Fat lines correspond to propagators of heavy particles, the rest to light particles. In Figs. 1b–1d various IR subgraphs are shown with dashed lines. Heavy knots in each case are clearly visible as connected components built of solid—normal and fat—lines.

5.3

Integrating δ-functions

Before substituting explicit expressions for E◦γ into (5.3), the dependence of each term in (5.3) on p should be studied. One can see that γ does not depend on those components of p that correspond to the loops of Γ\γ. Let us denote the collection of such momenta as pΓ\γ,int . Furthermore, each loop of Γ\γ belongs to only one heavy knot hi from the decomposition (5.4), so that: pΓ\γ,int = {pi,int }i .

(5.5)

The remaining momentum components of p form exactly the proper variables of γ introduced in [1] and denoted as pγ , so that p = (pγ , pΓ\γ,int ).

(5.6)

23


Note that Γ\γ is independent of those components of pγ that correspond to the loops of γ. Denote as pΓ\γ,ext those components of γ on which Γ\γ does depend. Substituting (5.3) into (5.2) and using the explicit expressions for E◦γ derived in [1], we can rewrite (5.2) as: As◦Γ(k, m, M) = Tm.k ◦Γ(k, m, M) +

XX Z

{

γ

dp′γ Pγ,α (p′γ ) γ(p′γ , k, m)

α

Z

dpγ

Z

dpΓ\γ,int δγ,α (pγ )

h

i

× Tm,k ◦ Γ\γ(pΓ\γ,ext , pΓ\γ,int , k, m, M) },

(5.7)

The polynomials Pα and the δ-functions δα in (5.7) form full dual sets (cf. [1]) so that the sum over α represents the Taylor expansion: X

Pγ,α (p′γ )

α

Z

dpγ δγ,α (pγ ) × f (pγ ) ≡ Tp′γ ◦f (p′γ ).

(5.8)

(It does not matter that the expression that plays the role of f in (5.7) depends not on all pγ but only on some of the components of pγ , namely, on pΓ\γ,ext .) Using (5.8) and renaming p′γ,ext → pγ,ext , we get: As◦Γ(k, m, M) = Tm,k ◦Γ(k, m, M) +

XZ

h

dpγ(pγ , k, m) × Tm,k ◦TpΓ\γ,ext ◦ Γ\γ(pΓ\γ,ext , pΓ\γ,int , k, m, M)

γ

i

.

(5.9)

Now recall the representation of Γ\γ in terms of heavy knots (5.4), (5.5). It is easy to see that the last line in (5.9) can be rewritten as Y

τ ◦hi (pi,int , pi,ext , k, m, M),

(5.10)

i

where pi,ext comprises those components of pΓ\γ,ext on which the heavy knot hi depends. The operation τ in (5.10) Taylor-expands in m and in all the momenta that are external with respect to the subgraph hi , i.e. both pi,ext and the corresponding components of k. It should be stressed that each factor in (5.10) is independent of the rest of the original diagram Γ. Therefore, we arrive at the following expression of the As-operation for integrated Feynman diagrams: As◦Γ = τ ◦Γ +

X

{hi }

Y i

!

τ ◦hi × [Γ/

Y

hi ].

(5.11)

i

The meaning of this formula is as follows. One enumerates all sets of pairwise nonintersecting heavy knots hi and contracts each heavy knot hi to a point, replacing it by a formal series in its external momenta which is obtained by Taylor-expanding the corresponding Feynman integral. To avoid confusion, it should be understood that the IR-counterterms E◦γ in (5.3) have transformed into the expression in the square brackets in (5.11), while the “counterterms” τ ◦hi in (5.11) have originated from the expression Γ\γ in (5.3). In other words, “counterterms” and “non-counterterms” have changed places in the transition from (5.3) to (5.11).

24


5.4

As-operation for integrals: the final form

Eq.(5.11) can be rewritten in the following form which is analogous to the representation (2.1) of the R-operation: As◦Γ =

X

{hi }

Y i

!

∆as ◦hi × [Γ/

Y

hi ]M =∞ .

(5.12)

i

The summation here runs over all sets of pairwise non-intersecting subgraphs hi of Γ such that each vertex of Γ belongs to one of hi . (A subgraph h is defined as a subset of vertices of Γ together with some lines of Γ which are connected by both ends to the vertices included into h. Subgraphs are non-intersecting if they have no common vertices and, consequently, no common lines.) One of such sets consists of a single subgraph h = Γ which corresponds to the first term τ ◦Γ on the r.h.s. of (5.11). The operation ∆as coincides with τ on heavy knots, is a unity operation on single vertices without loops, and returns zero otherwise. Setting M = ∞ is a formal expression of the fact that one should discard from the r.h.s. those terms in which some heavy lines remained outside all hi so that (5.11) could be restored from (5.12). There is no need to introduce special rules for heavy knots containing tadpoles made of light lines (cf. Fig. 2), our formalism takes them into account correctly. For practical purposes it is sufficient to note that such heavy knots are nullified by the operation τ due to the well-known property of the dimensional regularization to nullify momentum integrals without dimensional parameters. The combinatorial resemblance of (5.12) and the expression (2.1) for the R-operation is nearly complete except for the minor difference described above. However, this difference can be easily taken into account to make the reasoning of sects. 1–3 to be applicable to the As-operation. But first expansions of diagrams Γ other than 1PI should be considered.

5.5

As-operation for non-1PI diagrams

We have derived explicit expressions for the operation As on 1PI integrals. Remarkably, one can expand arbitrary—not necessarily 1PI—integrals using the same expression (5.12). For the operation As to be applicable to any graph Γ, its structure should satisfy certain a priori requirements. Let us first enumerate them. If Γ is disconnected, i.e. there are two subgraphs Γ1 and Γ2 such that Γ = Γ1 × Γ2 , then the expansion of Γ is a product of the expansions of Γ1 and Γ2 , i.e. As◦(Γ1 × Γ2 ) = (As◦Γ1 ) × (As◦Γ2 ).

(5.13)

If Γ falls into two parts connected by only one light line (the corresponding propagator is denoted as Dl ), i.e. Γ = Γ1 × Dl × Γ2 , then As◦(Γ1 × Dl × Γ2 ) = (As◦Γ1 ) × Dl × (As◦Γ2 ).

(5.14)

25


This is due to the fact that both the mass of this line and the momentum flowing through it (which is a combination of the light external momenta of Γ) are expansion parameters so that Dl is proportional to a negative power of κ and need not be expanded. If Γ consists of two subgraphs connected by one heavy line (the corresponding propagator is denoted as Dh ), i.e. Γ = Γ1 × Dh × Γ2 , then As◦(Γ1 × Dh × Γ2 ) = (As◦Γ1 ) × (Tk ◦Dh ) × (As◦Γ2 ).

(5.15)

(Recall that Tk Taylor-expands in k, and that the momentum flowing through Dh is a combination of the light external momenta of Γ.) Now, note that the definition (5.12) does not use 1PI-ness of Γ. Therefore, let us define the operation As by (5.12) on any graph Γ. Let us prove that the operation thus defined satisfies the above a priori criteria (5.13)–(5.15). Eq.(5.13) is true because each hi from (5.12) lies strictly within either Γ1 or Γ2 , so that each partition of Γ consists of a partition of Γ1 and a partition of Γ2 , and a sum over all partitions of Γ is equivalent to a sum over partitions of Γ1 times a sum over partitions of Γ2 . The same argument proves (5.14) because Dl cannot belong to any of hi without making it 1PI with respect to light lines, which implies that each hi with non-zero ∆as lies strictly within either Γ1 or Γ2 . To prove (5.15) we note that Dh must always belong to one of the hi . Denote this heavy knot as h. Then h = h′ × Dh × h′′ (where h′ and h′′ each consist of at least one vertex) and ∆as ◦(h′ × Dh × h′′ ) = (∆as ◦h′ ) × (Tk ◦Dh ) × (∆as ◦h′′ ). (5.16) Then one applies the same argument as above. An induction with respect to the number of 1PI components of Γ enables one to conclude that (5.12) remains correct on all Feynman diagrams.

5.6

Exponentiation of ∆as.

Now we are in a position to apply the combinatorial results of sects. 1–3 to (5.12). Let ϕ and Φ be the fields of the light and heavy particles that have the masses m and M, respectively. Let L = L(ϕ, Φ) be an interaction Lagrangian. Consider the generating functional of Green functions of the light fields < T exp[L(ϕ, Φ) + ϕJ] >0 . It is assumed that all the external momenta of the Green functions are light, i.e. O(κ). We ignore the UV divergences for a while, because we are now interested only in the formal global structure of the operation As; one can simply assume that there are no UV divergences at all in the model described by L. As was noted above, the combinatorial structures of the As- and R-operations are very similar. The similarity goes far enough to allow one to write down the following

26

5 HEAVY MASS EXPANSIONS analogue of (2.31): As ◦ < T exp[L(ϕ, Φ) + ϕJ] >0 = < T exp[Leff,bare (ϕ) + ϕJ] >0 , where

h

Leff,bare (ϕ) = ∆as ◦ TeL(ϕ,Φ) − 1

i

Φ=0

.

(5.17) (5.18)

Setting Φ to zero in (5.18) is formally equivalent to setting M = ∞ in (5.12) and is an expression of the fact that each heavy line in (5.11) must belong to one of the heavy knots. The algorithm of (5.18) is as follows: (i) all the diagrams contributing to T exp L(ϕ, Φ) are evaluated; (ii) ∆as nullifies all the terms except those which correspond to connected diagrams that are 1PI with respect to light lines; (iii) on single vertices, ∆as acts as a unit operation, i.e. ∆as ◦[L(ϕ, Φ)]Φ=0 = [L(ϕ, Φ)]Φ=0 ;

(5.19)

(iv) the remaining terms are exactly the heavy knots and can be represented as (2.28), where F I also depends on m and M. ∆as performs the formal Taylor expansion of F I with respect to m and all the external momenta of such terms, i.e. pi in (2.28). Note that the recipe for ∆as allows a remarkably “fool-proof” reformulation (which may be useful e.g. in writing computer programs): it is sufficient to apply the τ operation to all connected terms generated by the T -exponent. Then the terms that should be discarded will either develop singularities 1/0, or be nullified owing to the property of the dimensional regularization to nullify integrals without dimensional parameters like Z dD p p−2a = 0. It is no less remarkable that this “fool-proof” reformulation stays valid in the most general case comprising heavy external momenta with arbitrary linear restrictions.

5.7

Explicitly convergent form for the heavy mass expansion

As was noted in [1], our method of expansion leads to expressions containing spurious divergences in different terms which cancel in the final result. Let us reorganize our asymptotic expansions in such a way as to reexpress them in an explicitly convergent form. In (5.18), Leff,bare (ϕ) is a Lagrangian functional (in terms of subsect. 3.1) of the field ϕ whose coefficients (“coupling constants”) are functions of the heavy masses. Irrespective of the specific values of these functions, the diagrams on the r.h.s. of (5.17) may contain UV divergences. Using (3.11), one can explicitly extract the R-operation on the r.h.s. of (5.17) and rewrite it as: As◦ < T exp[L(ϕ, Φ) + ϕJ] >0 = R◦ < T exp[Leff (ϕ) + ϕJ] >0 ,

(5.20)

27

5 HEAVY MASS EXPANSIONS where Leff (ϕ) = ξ −1 [Leff,bare (ϕ)].

(5.21)

Since the mapping ξ −1 transforms a Lagrangian functional again into a Lagrangian functional, Leff (ϕ) can be represented as Leff (ϕ) =

XZ

dx geff,n (g, M, µ, ǫ)jn (x),

(5.22)

n

where jn (x) are defined in subsect. 3.1 and g denotes the set of coupling constants of the original interaction Lagrangian L; the “effective couplings” geff,n have the form: geff,n (g, M, µ, ǫ) =

∞ X µ l(2ǫ) 1 X k geff,n,l (ǫ), g M dn k=0 l≥0 M

(5.23)

where the summation over powers of g reminds one that we are working within perturbative framework; geff,n,l (ǫ) have a pole singularity at ǫ ∼ 0 of a finite order depending on n and l, while the integer exponents dn (some which for a few lowest values of n may be negative) are determined on dimensional grounds. Indeed, the coefficients geff,n are polynomial expressions built of UV counterterms (each of which is just a polynomial in 1/ǫ) and heavy knots which are Taylor-expanded in powers of their external momenta and m. The resulting loop integrals depend only ′ ′ on M, and on dimensional grounds this dependence has the form M d −2ǫl × c(ǫ) where d′ is an integer and l′ the corresponding loop number, while c(ǫ) has a pole singularity ′ 1/ǫl (the latter information is strictly speaking not necessary). The powers of momenta and the light masses m resulting from the above Taylor expansion are included into j (recall the definition in subsect. 3.1). Recalling that the UV renormalization parameter µ enters all expressions via integer powers of µ2ǫ , one arrives at (5.22) and (5.23). Note that the range of summation over l is finite for each n and k. Now one can easily convince oneself that the parameters of the effective Lagrangian are finite at ǫ = 0. Indeed, the initial expression (the l.h.s. of (5.20)) is finite at ǫ = 0 by construction in each order of the perturbation theory (in the absence of UV divergences in the initial model—recall the assumption at the beginning of this section). When transformed into the form of the r.h.s. of (5.20), it is, of course, still finite but has now the form of UV-renormalized (and therefore, finite) loop integrals times products of geff,n (g, M, µ, ǫ), in each order of the perturbation theory. Taking into account linear independence of jn , one concludes that all geff,n (g, M, µ, ǫ) are finite at ǫ = 0.

5.8

As-operation on UV-divergent diagrams

Eq.(5.2) was used to define the As-operation on UV-convergent diagrams. However, that definition can be formally extended to the UV-divergent case (then, of course, the UV divergences should be regularized; but the dimensional regularization which is used at the intermediate steps of the As-operation for products as derived in [1],

28


regularizes UV divergences as well; therefore, no new regulators are really needed). The combinatorial results (5.17)–(5.21) remain valid though now the effective couplings geff,n (g, M, µ, ǫ) are not finite at ǫ ∼ 0 because they inherit the UV divergences of the unrenormalized Green functions on the l.h.s. of (5.17). Let R◦Γ(k, m, M) be a renormalized graph (the dependence on the renormalization parameter µ is implicit; note that we are dealing here with integrated graphs, so that there is no dependence on the loop momenta). The R-operation was defined by (2.1), which can be represented as follows: R◦Γ(k, m, M) =

X

Zi (1/ǫ, m, M)Γi (k, m, M),

(5.24)

i

where i enumerates all the individual terms in the sum in (2.1), Zi are products of divergent coefficients of the corresponding UV counterterms and are polynomials of all their arguments, Γi are the (unrenormalized) diagrams obtained from Γ by shrinking the corresponding UV-subgraphs to points. Naively, to expand (5.24) one would expand each term on the r.h.s.: the factors Zi are polynomials in m and are already “expanded”, and Γi can be expanded by a straightforward application of the definition (5.2). Indeed, all the intricacies of the As-operation are essentially aimed at extracting the terms that are non-analytical in light parameters, and the very fact of the polynomial dependence of UV counterterms on masses and momenta makes one expect that UV renormalization in the MS-scheme and the As-expansion are, in a sense, “orthogonal” (cf. the analysis of [17]). Motivated by the above, we define the As-operation on renormalized graphs by applying the As-operation (5.2) termwise to Γi on the r.h.s. of (5.24) as follows: def

As◦R◦Γ(k, m, M) =

X

Zi (1/ǫ, m, M)As◦Γi (k, m, M).

(5.25)

i

We assert that the expansion thus obtained is a correct asymptotic expansion for R◦Γ(k, m, M). This is not obvious, but for that matter neither is obvious the UV finiteness of the r.h.s. of (5.24) or even (2.1). For a practically oriented reader it may be sufficient to learn the following. On the one hand, the final expressions for the coefficient functions of short-distance expansions (see subsect. 6.3) imply highly non-trivial cancellations of UV and IR divergences between different terms which cannot be understood without the underlying theory. Such cancellations were checked by explicit three-loop calculations in [11] and in several other calculations (see e.g. [18]). On the other hand, from the point of view of our methods there is no essential difference between the short-distance operator-product expansion and the most general Euclidean regimes with heavy masses, and all the nontrivial patterns of interaction of UV and IR divergences manifest themselves at the three-loop level. One should also take into account that we are dealing with relatively primitive although increasingly cumbersome integrals of rational functions where all the effects are governed by the power counting (this point is emphasized and exhibited in the regularization independent formalism developed in the parallel series of papers

29


[14]–IV), so that there is no room for pathologies. Therefore, the explicit calculations of [11] and [18] provide a sufficient evidence in favour of the above assumption. For a formally oriented reader we note that as was proved in [15], the R-operation can be represented in such a form that its UV finiteness becomes obvious. Moreover, the reasoning of [17] demonstrates that the proof of correctness of the above definition reduces to verifying commutativity of two As-operations—one corresponding to the expansion under consideration, the other being implicit in the construction of UV subtractions along the lines of [15]. Such commutativity is essentially due to commutativity of the corresponding formal expansions of the individual factors, and verification of the fact that it is indeed inherited by As-operations on the entire integrands is a rather straightforward technical excercise. An informal discussion of the proofs of [17] is [16]. In what follows, we will concentrate on the combinatorial structure of the definition (5.25). (Note that the aggregate operation As◦R defined in (5.25) was denoted as As and studied as a whole in our original publication [2], and the starting point there was the so-called EA-expansion. Such a way of reasoning resulted in superfluous complications which are avoided in the present exposition.)

5.9

As-operation in models with UV divergences

If the model described by the interaction Lagrangian L(ϕ, Φ) possesses UV divergences, then the UV-renormalized generating functional of Green functions can be represented as (cf. (2.33)): R◦ < T exp[L(ϕ, Φ) + ϕJ] >0 =< T exp[LR (ϕ, Φ) + ϕJ] >0 .

(5.26)

with LR (ϕ, Φ) defined analogously to (2.33). Since the As-operation is defined to act formally on each term of the renormalized diagrams, one can apply it to both sides of (5.26). Then one can simply use the results obtained for the case of the models without UV divergences (see (5.17)–(5.21)). Our final formulae are as follows: R◦ < T exp[L(ϕ, Φ) + ϕJ] >0

≃

As◦R◦ < T exp[L(ϕ, Φ) + ϕJ] >0

M →∞

≡ R◦R−1 ◦As◦R◦ < T exp[L(ϕ, Φ) + ϕJ] >0 ≡ R◦ < T exp[Leff (ϕ) + ϕJ] >0 , where

h

i

Leff,bare (ϕ) ◦ −1 , Leff (ϕ) = ∆−1 UV T e

h

Leff,bare (ϕ) = ∆as ◦ T eLR (ϕ,Φ) − 1 h

i

i

Φ=0

LR (ϕ, Φ) = ∆UV ◦ TeL(ϕ,Φ) − 1 .

,

(5.27) (5.28) (5.29) (5.30)

30


The algorithms of the three ∆-operations have been described in detail in sects. 1.1, 2.8, and 4.4. Eq.(5.27) means that the effects of virtual presence of heavy particles on the effective low energy theory of light particles can be described by an effective Lagrangian Leff (explicitly given by (5.28)–(5.30)) to all orders in 1/M. Our derivation of this result is valid within the MS-like renormalization schemes and in all models including those with massless particles like QCD. Note that the R-operations in the original model and in the effective model need not be the same; this can be equivalently described by saying that the renormalization parameters in the two models may be different. In such a case the parameters of Leff will depend on both renormalization parameters. It should also be recalled that the As-operation derived in [1] is essentially unique because it leads to series in powers and logarithms of the expansion parameters. This uniqueness property (up to the choice of renormalization schemes) is inherited by (5.27)–(5.30), whatever the guise in which such expansions might appear when derived by alternative methods. Our form of presentation is dictated by computational convenience. The property of uniqueness also facilitates the study of gauge properties of Leff (cf. [10]).

5.10

Example

To get a feeling of how the above formulae work, consider QED with a light electron ψ and a heavy muon Ψ. The well-known interaction Lagrangian is denoted as L(ψ, Ψ, A) where A is the photon field. Restricting ourselves to the two-photon sector in the 1-loop approximation, we get (fat lines correspond to the muon): h

LR (ψ, Ψ, A) = ∆UV ◦ T eLQED (ψ,Ψ,A) − 1 = ∆UV ◦

"

#

= δZ i

Z

+ ∆UV ◦

i

"

#

+ ...

1 2 dx − Fµν (x) + . . . , 4

h

Lbare,eff (ψ, A) = ∆as ◦ TeLR (ψ,A) − 1 = LR (ψ, A) + ∆as ◦ and, finally,

" h

#

(5.31)

i

ψ=0

+ ...,

Leff,bare (ψ,A) ◦ −1 Leff (ψ, A) = ∆−1 UV T e

= Leff,bare (ψ, A) +

◦ ∆−1 UV

"

#

(5.32)

i

+ ....

(5.33)

31

6 GENERALIZED OPERATOR-PRODUCT EXPANSIONS Since ◦ ∆−1 UV

"

#

= −∆UV ◦

"

#

,

(5.34)

we obtain: Leff (ψ, A) = ∆as ◦

"

#

+ ∆UV ◦

"

#

+ ...

1 2 = δz(M , µ )i dx − Fµν (x) + . . . , 4 where the dots denote other field structures, and 2

δz(M 2 , µ2 ) =

2

Z

i e2 h 2 2 c + c log(M /µ ) + O(1/M 2). 1 2 16π 2

(5.35)

(5.36)

Note that cancellation of the UV divergences in the final result is obvious even without explicit calculations.

6

Generalized operator-product expansions

In this section we turn to the case when the set of heavy parameters includes external momenta. Recall that we regard the diagrams and Green functions to be expanded as distributions with respect to the heavy external momenta [1]. We start by considering in subsects. 6.1–6.3 the case corresponding to the usual short-distance operator-product expansion, i.e. when there are no heavy masses and no linear restrictions on the heavy external momenta except the overall momentum conservation. In subsect. 6.1 notations are introduced and the As-operation for integrals is derived for the case under study, in subsect. 6.2 the global structure of the As-operation as applied to Green functions is studied, and in subsect. 6.3 the final results are presented. Modifications due to heavy masses are studied in subsect. 6.4. In subsect. 6.5 we impose the so-called “natural” linear restrictions on the heavy momenta, and in subsect. 6.6 the corresponding version of the As-operation and the expansions for Green functions are presented. In subsect. 6.7 the structure of the general formulae is explained with a simple example. In subsect. 6.8 we briefly discuss the contact terms in the obtained expansions. The section is concluded in subsect. 6.9 with a discussion of the most general linear restrictions imposed on heavy momenta; it is pointed out that the contact terms in such a case may contain the so-called paralocal operators.

6.1

As-operation and heavy momenta

Let us assume that all the particles of the model are light, but some of the external momenta of the Green function to be expanded are heavy.

6 GENERALIZED OPERATOR-PRODUCT EXPANSIONS

32

More precisely, let j numerate the heavy external lines of the diagram Γ, then Qj are the corresponding ingoing external momenta. There are also some light external momenta collectively denoted as k, which are of order O(κ) together with the masses m (recall that κ is our standard notation of the expansion parameter [1]). The fact that Q are heavy is formally expressed as Q = O(1) as κ → 0.

(6.1)

One cannot assume all Qj to be independent of κ because of the momentum conservation which reads: X X Q=− k = O(κ). (6.2) Therefore, let us introduce the “heavy” components of Q that are independent of κ :

Q can be represented as:

¯ j ≡ Qj |κ=0. Q

(6.3)

¯ j + qj , Qj = Q

(6.4)

where qj = O(κ) are linear combinations of k. The momentum conservation should hold separately for heavy and light components, so that: X ¯ j = 0. Q (6.5) j

¯ except (6.5). Considering We have assumed that there are no other restrictions on Q the diagram Γ as a distribution with respect to the heavy momenta means that we expand expressions of the form Γ(k, m, F ) ≡

Z

¯ ′ F (Q) ¯ dQ

Z

¯ + q, k, m), dp Γ(p, Q

(6.6)

where F is a smooth test function independent of κ, and the integration runs over the manifold described by (6.5). It is convenient to represent F as a vertex attached to the vertices corresponding to Q by the lines that are heavy by definition—see Fig. 3. To expand Γ(k, m, F ), one repeats the reasoning of subsects. 4.1–4.4 and arrives at the following equation instead of (5.12): As◦Γ =

X

(∆as ◦h) × [Γ/h].

(6.7)

h

Now there is only one heavy knot shrunk to the point in each term in the sum. Its description—owing to the above agreement that the lines connecting the F -vertex with Γ are heavy—coincides with that given in subsect. 5.2: it must be 1PI with respect to light lines. But it may be easier to follow the “fool-proof” recipe given at the end of subsect. 5.6, which remains valid here. A reasoning similar to that of subsect. 5.5 shows that the above formula is also valid in the case of a non-1PI graph Γ; in particular, it is valid for disconnected graphs.


6.2

33

As-operation on Green functions

Let us now turn to Green functions. Let Hj (x) be local products of light fields ϕ(x) and their derivatives. Consider the following generating functional of Green functions: nY

0 , H

j

(6.8)

where the tilde marks Fourier transforms and L is a local functional (see the definition in subsect. 3.1). To obtain specific correlators from (6.8), it is sufficient to perform suitable variations with respect to the coefficient functions of L and replace L by the Lagrangian L. The momenta corresponding to any additional operator insertions are light, i.e. O(κ), by definition. Without loss of generality, the test function is introduced as follows: o

n

G(F, L) ≡< T F ∗ HeL >0 , where F ∗H=

Z

¯ ′ F (Q) ¯ dQ

Y

˜ j (Q ¯ j + qj ), H

(6.9) (6.10)

j

¯ j and qj are the same as defined in subsect. 6.1. where the momenta Q The sum of all qj should in general not be taken to be zero in (6.9) in order to get rid of the disconnected diagrams contributing to (6.9) which normally are of no interest in phenomenological applications. If, however, the kinematics of the problem P require that qj = 0 for the connected component, one can take the corresponding limit termwise in the final expansion. Taking such limits commutes with the expansion procedure unless there are connected diagrams like the one shown in Fig. 4 where the wavy lines correspond to massless particles and may give rise to an infrared divergence. Whether or not the problem under study allows such terms can be checked by a straightforward inspection. However, it is still possible to regularize such contributions e.g. by introducing a non-zero mass (which, of course, should be considered as light in ¯ the expansion procedure); the Q-dependent part of the final expansion as obtained by our methods will be insensitive to the light mass structure of the model (see below). Now, using (6.7) and reasoning as in subsect. 5.6, we obtain the following analogue of (5.17): i o h n (6.11) As◦G(F, L) =< T ∆as ◦ TF ∗ HeL eL >0 .

The algorithm for evaluating ∆as [TF ∗ H exp L] is exactly the same as described in subsect. 5.6 (recall that according to the graphical conventions introduced after (6.6), the lines connecting the F -vertex with the proper Feynman diagram are heavy by definition; note also that the δ-function which expresses momentum conservation in the relevant diagrams contains a sum of only light momenta due to (6.5), so that a formal expansion in light parameters will not affect it; if one wishes, one could introduce a formal integration in some of the qj to get rid of it completely). One sees

34


that ∆as [TF ∗ H exp L] is a local functional in the sense of subsect. 3.1, so that (cf. subsect. 6.8): i h X bare CH,n (F, L)˜jn (q), (6.12) ∆as ◦ TF ∗ HeL = n

where q =

P

qj and

bare CH,n (F, L)

=

Z

¯ ′ F (Q)C ¯ bare (Q, ¯ L). dQ H,n

(6.13)

We have introduced the subscript “bare” to indicate that the C’s contain divergences, so that a procedure analogous to what was described in subsect. 5.7 is needed, in order to transform (6.11) to an explicitly convergent form. To accomplish this, one proceeds as follows: As◦G(F, L) =

X

n

o

bare CH,n (F, L) < T ˜jn (q)eL >0

n

δ < T eL >0 δL (q) n n X bare δ −1 = CH,n (F, L) R◦ < Teξ [L] >0 δLn (q) n =

=

X m

(

X

X

bare CH,n (F, L)

bare CH,n (F, L)Zn,m

n

)

−1 R◦ < T ˜jm (q)eξ [L] >0 ,

n

o

(6.14)

where we have used the fact that ξ −1[L] defined in sect. 3 is a local functional. To complete the reasoning, one should take into account that the initial expression on which the As-operation acts may contain UV divergences and be renormalized via the R-operation, as follows: n

o

GR (F, L) ≡ R◦ < T F ∗ HeL >0 .

(6.15)

The arguments justifying correctness of a straightforward application of As to renormalized expressions are the same as in subsect. 5.8.

6.3

Short-distance expansions

Now let us exhibit the general structure of the expressions for the case when L = L+ϕJ where L is a Lagrangian functional. Using the results of sect. 4 (cf. (4.9)), one represents (6.15) as: GR (F, L + ϕJ) =

X

o

n

< T F ∗ ZH,a ∗ Ja e[LR +ϕJ] >0 ,

a

(6.16)

where Ja are multilocal products of the currents jn defined in subsect. 3.1, Z is a “matrix” of divergent coefficients, and the ∗’s denote contractions over the heavy momenta. LR is defined in (2.26). Applying As to both sides and using (6.14), one gets: As◦GR (F, L + ϕJ) =

Z

¯ ′ F (Q) ¯ dQ

X m

n

o

¯ g, µ)R◦ < T ˜jm (q)e[L+ϕJ] >0 , cm (Q,

(6.17)

35

6 GENERALIZED OPERATOR-PRODUCT EXPANSIONS where

Z

¯ ′ F (Q)c ¯ m (Q, ¯ g, µ) ≡ dQ

XX a

n

CJbare (F ∗ ZH,a , LR )Zn,m a ,n

(6.18)

are linear functionals of the test function F that are finite in the limit ǫ → 0 (which is proved similarly to the reasoning at the end of subsect. 5.7); g are the coupling constants of the Lagrangian L, and µ is the renormalization parameter. The expansion (6.17) has the form of a familiar operator product expansion at short distances. It should be stressed that (6.18)—despite its cumbersome appearance—describes a fully constructive algorithm for getting explicit expressions for the coefficient func¯ g, µ). This algorithm is equivalent to the one described in [11], where an tions Cm (Q, assumption on existence and properties of the operator-product expansion in the MSscheme was made. Namely, in [11] it was assumed that the coefficient functions of such OPE are analytical in masses. One can easily see that the expressions (6.17)–(6.18) prove that assumption: indeed, the coefficient functions as defined above have turned out to be independent of m, but the local operators j have been allowed to contain non-negative integer powers of m; if the local operators are built of only fields, then the coefficient functions would become analytical in masses. Non-trivial examples of two- and three-loop calculations of coefficient functions using this algorithm together with explicit formulae can be found in [11] and [18]. Finally, note that if one wishes to consider Green functions of composite operators, then suitable variations in L should be introduced into (6.14), (6.15) etc., giving rise to additional terms in the final result. Thus, if one considers the expansion of a correlator of the form o n (6.19) < T ˜jn (Q)˜jm (−Q + q)˜jl (k)eL >0 , then the final expansion will contain the correlators n

o

< T ˜ja (q)˜jb (k)eL >0

and

n

o

< T ˜ja (q + k)eL >0 .

(6.20)

The latter expression is a typical “vacuum condensate”, i.e. a vacuum expectation value of a local operator (cf. [12]). It is not equal to zero, even within perturbation theory, because the normal ordering of local products of fields is not used in the MS-scheme.

6.4

Effects of heavy particles on operator expansion

If there are heavy masses in the model, then the above results will get modified in the following way. The expression (5.12) for the As-operation will remain valid provided one includes the heavy knots corresponding to the F -vertex that were described after (6.7). Note that the “fool-proof” recipe of subsect. 5.2 is still valid here. Instead of (6.11) one will have: n

h

i

h

As◦G(F, L) =< T ∆as ◦ TF ∗ HeL exp ∆as ◦ T eL − 1 And the final result (6.17) will take the form As◦GR (F, L(ϕ, Φ) + ϕJ)

io

>0 .

(6.21)

36

6 GENERALIZED OPERATOR-PRODUCT EXPANSIONS =

Z

¯ ′ F (Q) ¯ dQ

X

n

¯ M, g, µ, q) × R◦ < T jm ( cm (Q,

m

o

X

q) exp [Leff (ϕ) + ϕJ] >0 ,

(6.22) where Leff (ϕ) is defined in (5.21), while cm are analytical in q. Note that the operators jm are here built of the light fields only, while the currents entering into the multilocal operator H in the initial expression are allowed to contain heavy fields as well.

6.5

Natural restrictions on heavy momenta

Let us turn to the case when there are linear restrictions imposed on the heavy external momenta other than the overall momentum conservation. Of immediate phenomenological importance are the so-called natural restrictions [3], [1]. In terms of position space, one can describe them as follows: all the “heavy” operator insertions are arranged into several groups, the distances within each group tend to zero while the distances between groups stay finite (cf. Fig.5). More precise definitions in terms of momentum representation are presented below. Consider (6.8) and let the “heavy” operator insertions Hj (which are allowed to be built of both heavy and light fields) be divided into several non-intersecting groups (numerated by λ) in such a way that the heavy momentum conservation holds separately within each group: X Qj = O(κ), for each λ. (6.23) j∈λ

¯: In terms of the independent momenta Q X

¯ j = 0, Q

for each λ.

(6.24)

j∈λ

¯ λ ) (where Q ¯λ For each group it is natural to introduce a separate test function Fλ (Q ′ ¯ j for j ∈ λ), and the corresponding integration dQ ¯ over the denotes the set of all Q λ manifold described by (6.24). Now eqs.(6.9) and (6.10) are replaced by: G(Fλ , L) ≡< T where Fλ ∗ Hλ =

Z

(

Y

L(ϕ,Φ)

>0 ,

(6.25)

˜ j (Q ¯ j + qj ). H

(6.26)

Fλ ∗ Hλ e

λ

¯ ′ Fλ (Q ¯ λ) dQ λ

Y

)

j∈λ

To use graphical representation, one introduces an F -vertex of the same type as described after (6.6) for each of the test functions Fλ .

6.6

Generalized operator expansions

The As-operation on a diagram Γ contributing to (6.25) will have the form As◦Γ =

X

{hj }

Y i

!

"

∆as ◦hi × Γ/

Y i

hi

#

M =∞,F =0

(6.27)

37


(cf. (5.12)), where the heavy knots h are 1PI with respect to light lines and must contain at least one F -vertex or heavy line. Setting M = ∞ and F = 0 is an expression of the fact that all the “heavy” elements (F -vertices and heavy lines) must be contained within the h’s. Applying As to (6.25), one obtains the following expression instead of (6.21): As◦G(Fλ , L(ϕ, Φ)) =

X

< T{

{Λ}

Y

Λ∈{Λ}



∆as ◦ T

Y

λ∈Λ



h

i

Fλ ∗ Hλ eL(ϕ,Φ)  × exp ∆as ◦ TeL(ϕ,Φ) − 1 } >0 ,

(6.28)

where {Λ} denotes a splitting of the set of all λ into non-intersecting subsets numerated by Λ, so that Y Y Y Fλ ∗ Hλ = Fλ ∗ Hλ , (6.29) Λ∈{Λ} λ∈Λ

λ

and the first summation on the r.h.s. of (6.28) runs over all such splittings {Λ}. The action of ∆as can be most easily described using the “fool-proof” recipe presented in subsect. 5.2: ∆as nullifies unconnected diagrams as well as all those connected ones which result in meaningless expressions like 1/0 when formally Taylor-expanded in light parameters. The remaining diagrams are exactly 1PI with respect to light lines and get formally Taylor-expanded in the light parameters. Each ∆as results in a local functional (cf. (6.12)), so that (6.28) can be represented as As◦G({Fλ }, L(ϕ, Φ)) =

X

< T{

{Λ}

Y

Λ∈{Λ}

 

X aΛ



CΛ,aΛ ,bare ({Fλ }λ∈Λ , M, L, {qj }j∈Λ , ǫ)˜jaΛ 

X

j∈Λ

× exp Leff,bare (ϕ)} >0 ,



qj 

(6.30)

where each C is a coefficient function which is independent of fields and is at the same time a linear functional with respect to each Fλ . The dependence on qj is analytical. To obtain an expansion for the renormalized Green functions in an explicitly convergent form one should—as has been done in all the special cases considered above— “sandwich” the As-operation between R and R−1 (cf. (5.27)). Further reasoning is similar to that of subsects.4.7 and 5.2–3: one should first get rid of the R-operation using the generalized Zimmermann identities of subsect. 4.1, then apply the As-operation, and, finally, use the identities of subsect. 4.2 to extract the R-operation and thus transform the expansion to an explicitly convergent form. In this way one arrives at the following final result: As◦R◦G({Fλ }, L(ϕ, Φ) + ϕJ) =

X

{Λ}

R◦ < T{

Y

Λ∈{Λ}

 X  cΛ,a

Λ

aΛ



({Fλ }λ∈Λ , M, g, {qj }j∈Λ ) ˜jaΛ 

× exp [Leff (ϕ) + ϕJ]} >0 ,

X

j∈Λ



qj 

(6.31)

38


which should be compared with (6.17); Leff is given by (5.21) and has the form of (5.22).

6.7

Example

To clarify the above general expansion, consider the case of two groups of two currents each (the corresponding phenomenological problem is the deep inelastic scattering of ¯ 1 and Q ¯ 2 , and two two deeply virtual photons [13]). There are two heavy momenta Q ¯ 1 ) and FCD (Q ¯ 2 ). The expression to be expanded is: test functions, FAB (Q Z

Z

¯ 1 dQ ¯ 2 FAB (Q ¯ 1 ) FCD (Q ¯ 2) dQ

n

o

¯ 1 )˜jB (−Q ¯ 1 + q1 )˜jC (Q ¯ 2 )˜jD (−Q ¯ 2 + q2 )e[L(ϕ,Φ)+ϕJ] >0 , × R◦ < T ˜jA (Q

(6.32)

where q are introduced to get rid of the non-connected components. Its expansion is as follows. There are two sets {Λ} over which the summation in (6.31) runs: {(A, B), (C, D)} and {A, B, C, D}. The first of them corresponds to that term in the expansion where the two pairs of currents are “shrunk” into (linear combinations of) local operators separately; the second corresponds to a single sum of local operators. The resulting expansion reads (cf. Fig.6): Z Z X X

×

a

¯ 1 dQ ¯ 2 FAB (Q ¯ 1 )FCD (Q ¯ 2) dQ

¯ 1 , M, g, µ)cCD,b (Q ¯ 2 , M, g, µ) cAB,a (Q

b

n

o

×R◦ < T ˜ja (q1 )˜jb (q2 ) exp[Leff (ϕ) + ϕJ] >0 +

X

n

a

6.8

o

¯1, Q ¯ 2 , M, g, µ)R◦ < T ˜ja (q1 + q2 ) exp[Leff (ϕ) + ϕJ] >0 . cABCD,a (Q

(6.33)

Contact terms in operator expansions

¯ For The expansions that we have derived are valid for arbitrary test functions F (Q). ¯ 1 and Q ¯ 2 , then the expansion example, if there are two independent heavy momenta, Q ¯ ¯ ¯ ¯ 2 ), δ(Q ¯ 1 )δ(Q ¯ 2 ) etc. will in general contain terms proportional to δ(Q1 ), δ(Q2 ), δ(Q1 + Q (for a discussion of the role of such terms see [1]). Let us explain how such terms are generated by the As-operation. Within the context of subsect. 6.2 (no heavy masses etc.), consider the case corresponding to the product of two currents in (6.8): F ∗H=

Z

¯ F (Q)H( ¯ ¯ + q)H(−Q). ¯ dQ Q

(6.34)

39

6 GENERALIZED OPERATOR-PRODUCT EXPANSIONS The non-trivial terms are generated from the following expression (cf. (6.12)): i

h

∆as ◦ TF ∗ HeL =

X

bare CH,n (F, L)˜jn (q),

(6.35)

n

where L is the interaction Lagrangian. There are two classes of terms generated by TF ∗HeL that give non-zero result after application of ∆as . The first class contains the terms whose graphs remain connected after the F -vertex is deleted. Such terms can be represented as: µ−2ǫ

Z

¯ F (Q)f ¯ (Q, ¯ p1 , . . . , pn , m, µ) × δ(q + dD Q

X

pj )

(ϕ(p ˜ j ) dD pj ),

Y

(6.36)

j

where the Feynman amplitude f depends on the masses m of the theory, contains one factor µ2ǫ per each loop, and the δ-function that expresses the momentum conservation is shown explicitly. The action of ∆as on (6.36) consists in Taylor-expanding f in masses and pj . The result has the form: ∆as (6.36) = ◦



× µ−2ǫ

Z

Z

¯ (Q) ¯ dD QF

X

¯ µ) fa (Q,

a

Pa (m, p1 , . . . , pn )δ(q +

X

pj )

j



(ϕ(p ˜ j )dD pj ) .

Y

(6.37)

The square-bracketed term in (6.37) corresponds to ˜j in (6.12), and the dependence of ¯ µ) on Q ¯ and µ is as follows: fa (Q, ¯ µ) = f¯a (Q, ¯ fa (Q,

m2 Q2

ǫ

).

(6.38)

In the l-loop approximation, f¯a is a polynomial of order l in its second parameter, while its dependence on the first parameter is determined by power counting and covariance properties. The second class contains the heavy knots that consist of two parts connected only via the F -vertex (cf. Fig. 7). If there are no heavy masses, then there is only one such subgraph that is not nullified by the operation τ —it consists of the two vertices corresponding to the “heavy” operators H (see (6.34)) connected to the F -vertex. The corresponding contribution to (6.35) has the following form (for definiteness we assume that H(x) = ϕ(x)2 ): Z

∆as [ ◦

dQ F (Q) δ(q + Q + p1 + p2 )δ(−Q + p′1 + p′2 ) ×ϕ(p ˜ 1 )ϕ(p ˜ 2 )ϕ(p ˜ ′1 )ϕ(p ˜ ′2 ) dp′1 . . . dp′2 ]

= τ [dQ F (Q) δ(−Q + p′1 + p′2 )] × δ(q + p1 + p2 + p′1 + p′2 )ϕ(p ˜ 1 )ϕ(p ˜ 2 )ϕ(p ˜ ′1 )ϕ(p ˜ ′2 ) dp1 . . . dp′2 ,

(6.39)

40


where the two δ-functions in the first line are inherited from the two operators H (cf. (3.1)), and we have extracted a δ-function in the last line that depends only on the light momenta. We have deliberately not performed the integration over Q on the r.h.s. because τ should Taylor-expand the square-bracketed expression in p′i and this can be done by applying τ directly to the δ-function. The result has the form XZ

dQ [F (Q)δ (α) (Q)] × jα (q),

(6.40)

a

and one sees that the “bare” coefficient functions in (6.12) receive δ-functional contributions. Such contributions—“dressed” by the appropriate divergent renormalization factors (cf. (6.18))—are finally inherited by the coefficient functions of the short-distance expansion (6.17). It may be not quite obvious that the coefficient functions thus constructed will be integrable around Q ∼ 0. However, cancellations between different terms can be traced if one performs the expansion procedure explicitly starting from the As-operation for products of singular functions prior to performing the integrations of δ-functions described in subsect. 5.3. Alternatively, an interested reader can verify validity of our recipes by straightforward calculations in a simplest situation, e.g. within the model ϕ3 in two dimensions for the currents H(x) = ϕ2 (x) in one loop approximation.

6.9

General linear restrictions on heavy momenta and paralocal operators

Consider the following operator product instead of (6.10): F ∗H=

Z



¯ ′ F (Q) ¯ δ(q + dQ

X

¯j) Q

j

Y

δ(qσ +

X j

λ



¯ j )  cλ,j Q

Y j



˜ j (Q ¯ j + qj ) , (6.41) H

where the δ-functions are introduced to impose linear restrictions on the heavy momenta. With a special choice of cλ,j one reproduces the restrictions (6.24). Examples of more general restrictions are presented in Fig. 8. We are not aware of any phenomenological applications where such restrictions might emerge naturally. Therefore, we offer only a few comments concerning the most general case (6.41).

First, it should be stressed that the expansion procedure as described by (6.11) remains fully correct and well-defined provided the “fool-proof” recipe for ∆as from subsect. 5.6 is used. No simple graphical description for heavy knots exists, though. Second, the representation (6.12) should be replaced by a more general one: the sum on the r.h.s. may now contain operators that can not be interpreted as local operators and differ from (3.1) by additional δ-functions besides the one expressing momentum conservation. Such operator monomials can be called paralocal operators [3]. A simple example of a paralocal operator in position-space representation may be as follows: Z Y X Y P(z) = ξλcλj ), (6.42) dξλ ϕ(z + λ

j

λ


41

but more complicated patterns are possible: note in this respect that a natural positionspace representation for the multilocal operators which are a special case of paralocal ones, is J1 (z1 )..Jn (zn ), (6.43) where all J are local operators. The renormalization properties of paralocal operators can be studied most easily using the techniques of [15] (also [16]) and are similar to those of (multi-) local ones: a paralocal operator renormalizes via paralocal ones. Zimmermann identities (including the inverted ones) can also be generalized to this case. Therefore, expansions in explicitly convergent form for the case of general non-natural restrictions on heavy momenta can be obtained in a manner completely similar to the case of natural restrictions.

Conclusions We have developed a simple combinatorial technique for studying global exponentiation properties of the As-operation and obtained Euclidean asymptotic expansions for MS-renormalized Green functions of arbitrary local operators in arbitrary models. The expansions are true infinite asymptotic series that run in powers and logs of the expansion parameter. The obtained expansions exhibit perfect factorization of heavy and light dimensional parameters (masses and external momenta) which means e.g. that the coefficient functions of operator expansions are analytical in light masses and momenta. It should be stressed that the expansions are valid in models with massless particles like QCD, and are most convenient for practical calculations. Uniqueness of such expansions (cf. [1]) greatly facilitates study of their properties in gauge models.

Acknowledgments. The authors thank A. N. Kuznetsov for a critical reading of early versions of the manuscript. One of the authors (F. T.) is indebted to A. V. Radyushkin for his continuous stimulating interest in this work, to W. A. Bardeen and R. K. Ellis for the hospitality at the Theory division of Fermilab where the final version of the manuscript was completed, and to J. C. Collins for a careful reading of the manuscript and supplying a list of misprints and comments.

42

References [1] F. V. Tkachov: “Euclidean asymptotic expansions of Green functions of quantum fields. (I) Expansions of products of singular functions”, preprint FERMILABPUB-91/347-T. [2] G. B. Pivovarov and F. V. Tkachov: “Combinatorics of Euclidean asymptotic expansions of Green functions. (IV)”, report P-0370, Inst. for Nucl. Research of the USSR Acad. Sci., Moscow, 1984. [3] G. B. Pivovarov and F. V. Tkachov: “The general form of Euclidean asymptotic expansions of Green functions. (V)”, report P-0459, Inst. for Nucl. Research of the USSR Acad. Sci.: Moscow, 1986; “Theory of Euclidean Asymptotic Expansions. Summary of results”, in: “Quarks-86. Proceedings of the Seminar. Tbilisi, USSR”, pp.227-234. Eds.: A. N. Tavkhelidze et al., VNU Science Press: Utrecht, 1987. [4] G. B. Pivovarov and F. V. Tkachov: Teor. Mat. Fiz. 77 (1988) 51. [5] J. C. Collins: “Renormalization”, Cambridge Univ. Press: Cambridge, 1984. [6] O. I. Zavialov: “Renormalized Feynman diagrams”, Nauka: Moscow, 1979 (in Russian). [7] A. N. Vassiliev: “Functional Methods in Quantum Field Theory and Statistical Mechanics”, L.S.U.: Leningrad, 1976 (in Russian). [8] N. N. Bogoliubov and D. V. Shirkov: “Introduction to the Theory of Quantized Fields”, Nauka: Moscow, 1984. [9] W. Zimmermann: “Local operator products and renormalization in quantum field theory”, in: “Lectures on elementary particles and quantum field theory”. Eds.: S. Deser et al. MIT: Cambridge (Mass.), 1970. [10] G. B. Pivovarov: Teor. Mat. Fiz. 80 (1989) 47. [11] F. V. Tkachov: Phys. Lett. 124B (1983) 212; S. G. Gorishny, S. A. Larin and F. V. Tkachov: Phys. Lett. 124B (1983) 217. [12] M. A. Shifman, V. I. Vainshtein and V. I. Zakharov: Nucl. Phys. B147 (1979)385. [13] I. I. Balitsky: Phys. Lett. 114B (1982) 52. [14] F. V. Tkachov and V. V. Vlasov: “Multiloop Feynman diagrams and distribution theory”, preprints McGill/91-03, 04; A. N. Kuznetsov, F. V. Tkachov and V. V. Vlasov: “Techniques of distributions in perturbative quantum field theory. (I) Euclidean As-operation for products of singular functions”, preprint PSU-T92/108 (to be published).

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[15] A. N. Kuznetsov and F. V. Tkachov: “MS-Scheme without dimensional regularization”, talk at “Quarks-88. Int. Seminar”, Tbilisi, USSR, 1988; “Multiloop Feynman diagrams and distribution theory. (III) UV renormalization as subtraction of asymptotics in momentum representation”, preprint NIKHEF/90-17, Amsterdam; Talk at “Renormalization-91”, Dubna: JINR, 1991. [16] A. N. Kuznetsov and F. V. Tkachov: “Euclidean asymptotic expansions of Green functions of quantum fields. (III) Aspects of UV renormalization” (in preparation). [17] A. N. Kuznetsov, F. V. Tkachov and V. V. Vlasov: “Multiloop Feynman diagrams and distribution theory. (IV) Asymptotic expansions of renormalized diagrams”, preprint FERMILAB-PUB-91/346-T. [18] S. A. Larin, F. V. Tkachov, and J. A. M. Vermaseren: Phys.Rev.Lett. 66 (1991) 862; S. A. Larin and J. A. M. Vermaseren: “The O(αS3 ) corrections to the Bjorken sum rule for polarized electroproduction and to the Gross-Llewellyn Smith sum rule”, preprint NIKHEF-H/90-23, Amsterdam, 1991. [19] C. H. Llewellyn Smith and J. P. de Vries: Nucl. Phys. B296 (1988) 991. [20] S. Gorishny: Nucl.Phys. B319 (1989) 633. [21] V. A. Smirnov: Commun. Math. Phys. 134 (1990) 109. [22] K. G. Chetyrkin: preprint MPI-Ph/PTh 13/91. [23] F. V. Tkachov: “Mathematical methods of perturbative QFT: a new paradigm”, in: Proc. Int. Seminar QUARKS-90 (Telavi, USSR, May 1990), World Scientific: Singapore, 1991; preprint McGill/91-05. [24] G. Bonneau: Nucl. Phys. B167 (1982) 261.

44

Figure captions Fig. 1. A diagram (a) and its heavy knots (a)–(d). All the external momenta are light. The fat lines correspond to propagators of heavy particles. Various IR-subgraphs are shown with dashed lines in (b)–(d). Fig. 2. (a) An example of a diagram with a ”light” tadpole subgraph (the notations are the same as in Fig. 1). The tadpole corresponds to a non-expandable global factor. The As-operation should not affect such factors. (b) and (c) are two IR-subgraphs. The contribution corresponding to the IR-subgraph shown in (b) is automatically set to zero by the operation τ due to the properties of dimensional regularization, so that the tadpole can be factored out in the result produced by the As-operation, as expected. Fig. 3. A graphical representation of the test function corresponding to the heavy external momenta. The lines connecting the F -vertex with the rest of the diagram are heavy by definition. Fig. 4. If the wavy lines correspond to massless scalar particles then setting q = 0 would result in an IR divergence in D = 4. Fig. 5. An example of kinematics of heavy momenta with linear restrictions. (a) differs from (b) by the restriction Q3 = −Q2 . (c) shows the corresponding position space picture: ξ, ζ → 0. Fig. 6. A diagrammatic illustration of (6.33). The two terms on the r.h.s. correspond to two ways of cutting off the heavy knots corresponding to the heavy external momenta. Fig. 7. Graphical representation of the heavy knots contributing to contact terms. Each blob corresponds to a subgraph that is 1PI with respect to light lines. If there are no heavy masses then each blob is just 1PI and is nullified by τ unless it consists of a single vertex. Fig. 8. Examples of “non-natural” restrictions on heavy momenta. (a) differs from Fig. 6 by an additional restriction Q1 = Q2 .

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